Exercise $2.$ Consider a small open economy with $2$ periods. Households receive endowment Y$2$ in period $2$ and are taxed T$2($lump$-$sum$)$ by the government. Households don$$t have any savings or borrowing. The government has an outstanding stock of government debt B $=1\mathrm{.}5$ issued in period $1,$ which implies repaying B$(1+$ r$)$ in period $2($if the government decides to repay$).$ Suppose this debt is held by risk$-$neutral foreign investors that can access borrowing$/$lending at the risk$-$free international interest rate r $=10\%.$ The government is benevolent and chooses to repay or default $($entirely$)$ on its debt to maximize consumption in period $2$ for the households. If the government defaults, the available endowment for consumption is given by Y$2(1$c$),$ where c $=13$ is the proportional cost of default. From the perspective of period $1,$ Y$2$ is a random variable that can take the value of Y$2=10$ with probability $0\mathrm{.}5$ and Y$2=6$ with probability $0\mathrm{.}5.$